Finite Group Databasebeta

← #19148685

Subgroups — #19148685

Order 5504.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14583627C213yesmaximal, complemented
2#14583625C211yesmaximal, characteristic, complemented
3#14583617C213yesmaximal, complemented
4#14583590C213yesmaximal, complemented
5#14583588C213yesmaximal, complemented
6Q8×D172C211yesmaximal, characteristic, complemented
7Q8×C22⋊D43C211yesmaximal, characteristic, complemented
8Q8×C22⋊D43C211yesmaximal, characteristic, complemented
9#14583511C213yesmaximal, complemented
10#14583507C213yesmaximal, complemented
11#14583453C213yesmaximal, complemented
12#14583427C213yesmaximal
13#14583375C211yesmaximal, characteristic, complemented
14C4×Q8×D43C211yesmaximal, characteristic, complemented
15C43×#519C211yesmaximal, characteristic, complemented
16C2×Q8×D43C2211yescharacteristic
17C2×Q8×D43C2211yescharacteristic
18#13063325C2213yes—
19#13063324C2213yes—
20#13063323C2213yes—
21#13063323C2213yes—
22#13063320C2213yes—
23#13063319C2216yes—
24#13063318C2216yes—
25#13063318C2213yes—
26#13063317C2216yes—
27#13063317C2213yes—
28#13063316C2213yes—
29#13063315C2216yes—
30#13063315C2213yes—
31#13063315C2216yes—
32#13063311C2213yes—
33#13063310C2213yes—
34#13063307C2213yes—
35#13063305C2213yes—
36#13063305C2213yes—
37#13063304C2213yes—
38#13063304C2213yes—
39#13063303C2213yes—
40#13063301C2213yes—
41#13063301C2213yes—
42#13063298C2211yescharacteristic
43Q8×Dic43C2211yescharacteristic
44Q8×Dic43C2211yescharacteristic
45C4×Dic86C2216yes—
46C4×Dic86C2213yes—
47C4×Dic86C2213yes—
48C4×Dic86C2213yes—
49#13063290C2213yes—
50#13063290C2213yes—
51C4×D172C2213yes—
52C4×C22⋊D43C2213yes—
53C4×C22⋊D43C2213yes—
54#13063283C2213yes—
55#13063282C2216yes—
56#13063282C2213yes—
57C4⋊C4×D43C2213yes—
58C42×D43C2213yes—
59C22×Q8×C43C2211yescharacteristic
60C4×Q8×C43C2211yescharacteristic
61C4×Q8×C43C2213yes—
62C4×Q8×C43C2211yescharacteristic
63C4×Q8×C43C2213yes—
64#117×C43C2213yes—
65#117×C43C2213yes—
66C2×Q8×C43—24no—
67C2×Q8×C43C2313yes—
68C2×Q8×C43C2311yescharacteristic
69C22×C4×C43C2313yes—
70C2×C22⋊D43C2311yescharacteristic
71C2×C22⋊D43C2311yescharacteristic
72Q8×D43—24no—
73Q8×D43—24no—
74C2×D172C2311yescharacteristic
75C2×C4×D43C2311yescharacteristic
76C2×C4×D43C2313yes—
77C2×C4×D43C2313yes—
78C2×C4×D43C2313yes—
79C2×Dic86C2313yes—
80C2×Dic86C2313yes—
81C2×Dic86C2313yes—
82C4⋊C4×C43C2313yes—
83C4⋊C4×C43C2313yes—
84C4⋊C4×C43C2313yes—
85C4⋊C4×C43C2313yes—
86C22⋊C4×C43C2311yescharacteristic
87C22⋊C4×C43C2313yes—
88C42×C43C2313yes—
89C42×C43C2316yes—
90C42×C43C2313yes—
91C22⋊Dic43C2313yes—
92C22⋊Dic43C2313yes—
93(C2×D43)⋊C4C2311yescharacteristic
94(C2×D43)⋊C4C2313yes—
95(C2×D43)⋊C4C2313yes—
96(C2×D43)⋊C4C2313yes—
97(C2×D43)⋊C4C2313yes—
98(C2×D43)⋊C4C2313yes—
99C4⋊Dic43C2313yes—
100C4⋊Dic43C2313yes—
101C4⋊Dic43C2313yes—
102C4⋊Dic43C2316yes—
103C4⋊Dic43C2313yes—
104Dic43⋊C4C2316yes—
105Dic43⋊C4C2313yes—
106Dic43⋊C4C2313yes—
107Dic43⋊C4C2316yes—
108Dic43⋊C4C2316yes—
109Dic43⋊C4C2313yes—
110Dic43⋊C4C2313yes—
111Dic43⋊C4C2311yescharacteristic
112Dic43⋊C4C2316yes—
113Dic43⋊C4C2313yes—
114C4×Dic43C2313yes—
115C4×Dic43C2313yes—
116C4×Dic43C2313yes—
117C4×Dic43C2316yes—
118C4×Dic43C2313yes—
119C23×C43C2411yescharacteristic
120C22×D43C2411yescharacteristic
121C22×D43C2411yescharacteristic
122Q8×C43—26no—
123Q8×C43#4114yes—
124C2×C4×C43—26no—
125C2×C4×C43C2413yes—
126C2×C4×C43C2411yescharacteristic
127C2×C4×C43C2413yes—
128C2×C4×C43C2411yescharacteristic
129C2×C4×C43C2413yes—
130C2×C4×C43C2413yes—
131C22⋊D43—22no—
132C22⋊D43—22no—
133C2×Dic43C2411yescharacteristic
134C2×Dic43C2413yes—
135C2×Dic43C2413yes—
136C2×Dic43C2413yes—
137C2×Dic43C2413yes—
138C2×Dic43C2411yescharacteristic
139D172—22no—
140C4×D43—26no—
141C4×D43—26no—
142C4×D43—26no—
143C4×D43—22no—
144Dic86—26no—
145Dic86—26no—
146Dic86—26no—
147C22×C43—21no—
148C22×C43—21no—
149C22×C43—21no—
150C22×C43C2511yescharacteristic
151C2×D43—21no—
152C2×D43—21no—
153C2×D43—21no—
154C2×D43—21no—
155C2×D43—21no—
156C2×D432−1+412yes—
157C4×C43—23no—
158C4×C43—23no—
159C4×C43—23no—
160C4×C43—21no—
161C4×C43—21no—
162C4×C43C2×#4116yes—
163Dic43—23no—
164Dic43—23no—
165Dic43—23no—
166Dic43—23no—
167Dic43—21no—
168Dic432−1+412yes—
169#3523—431nomaximal, Sylow
170C2×C43—41no—
171C2×C43C22×#4111yescharacteristic
172C2×C43C2×2−1+411yescharacteristic
173C2×C43C2×2−1+411yescharacteristic
174D43—22no—
175D43—41no—
176#557—431no—
177#557—433no—
178#556—433no—
179#552—433no—
180#552—433no—
181D4×Q8—431no—
182D4×Q8—431no—
183D4×Q8—431no—
184#541—433no—
185#541—433no—
186#533—433no—
187#529—433no—
188#519—431no—
189#519D4311yescharacteristic, complemented
190C2×C4×Q8—431no—
191C43#352311yescharacteristic, complemented, Sylow
192C22×Q8—431no—
193C22×Q8—431no—
194C22×Q8C2×D4311yescharacteristic
195C4⋊Q8—433no—
196C4⋊Q8—433no—
197C4⋊Q8—433no—
198C4⋊Q8—433no—
199#126—433no—
200#126—433no—
201#126—436no—
202#126—436no—
203#125—433no—
204#125—436no—
205#125—436no—
206#125—433no—
207#125—436no—
208#125—433no—
209#124—433no—
210#123—433no—
211#123—433no—
212C22⋊Q8—433no—
213C22⋊Q8—433no—
214C22⋊Q8—433no—
215C22⋊Q8—433no—
216C22⋊Q8—433no—
217C22⋊Q8—433no—
218C22⋊Q8—433no—
219#121—431no—
220C4×Q8—431no—
221C4×Q8—431no—
222C4×Q8—433no—
223C4×Q8—433no—
224C4×Q8—433no—
225C4×Q8—433no—
226C4×Q8—433no—
227C4×Q8—436no—
228C4×Q8C2×D4311yescharacteristic
229C4×Q8C2×D4313yes—
230C4×Q8C2×D4311yescharacteristic
231C4×Q8C2×D4313yes—
232C4×D4—433no—
233C4×D4—433no—
234C4×D4—433no—
235#117—433no—
236#117—433no—
237#117—436no—
238#117C2×D4313yes—
239#117C2×D4313yes—
240C2×C4⋊C4—433no—
241C2×C42—433no—
242C2×Q8—24no—
243C2×Q8—433no—
244C2×Q8—433no—
245C2×Q8—433no—
246C2×Q8—864no—
247C2×Q8—864no—
248C2×Q8C22×D4313yes—
249C2×Q8C22×D4311yescharacteristic
250C2×D4—431no—
251C2×D4—431no—
252C2×D4—431no—
253C22×C4—431no—
254C22×C4—433no—
255C22×C4—433no—
256C22×C4—433no—
257C22×C4C22×D4313yes—
258C4⋊C4—433no—
259C4⋊C4—436no—
260C4⋊C4—433no—
261C4⋊C4—433no—
262C4⋊C4—433no—
263C4⋊C4—436no—
264C4⋊C4—436no—
265C4⋊C4—433no—
266C4⋊C4—433no—
267C4⋊C4—433no—
268C4⋊C4—431no—
269C4⋊C4—436no—
270C4⋊C4—436no—
271C4⋊C4—433no—
272C4⋊C4—433no—
273C4⋊C4C22×D4313yes—
274C4⋊C4C22×D4313yes—
275C4⋊C4C22×D4313yes—
276C4⋊C4C22×D4313yes—
277C22⋊C4—433no—
278C22⋊C4—433no—
279C22⋊C4—431no—
280C22⋊C4—433no—
281C22⋊C4—433no—
282C22⋊C4—433no—
283C22⋊C4—433no—
284C22⋊C4—433no—
285C22⋊C4C22×D4311yescharacteristic
286C22⋊C4C22×D4313yes—
287C42—433no—
288C42—433no—
289C42—433no—
290C42—436no—
291C42—433no—
292C42C22×D4313yes—
293C42C22×D4316yes—
294C42C22×D4313yes—
295C23—431no—
296C23—431no—
297C23C23×D4311yescharacteristic
298Q8—26no—
299Q8—866no—
300Q8—866no—
301Q8—866no—
302Q8#1062189814yes—
303D4—862no—
304D4—862no—
305D4—862no—
306C2×C4—26no—
307C2×C4—431no—
308C2×C4—433no—
309C2×C4—431no—
310C2×C4—433no—
311C2×C4—433no—
312C2×C4—433no—
313C2×C4—862no—
314C2×C4—866no—
315C2×C4—866no—
316C2×C4—866no—
317C2×C4C23×D4313yes—
318C2×C4C23×D4311yescharacteristic
319C2×C4C23×D4313yes—
320C2×C4C23×D4311yescharacteristic
321C2×C4C23×D4313yes—
322C2×C4C23×D4313yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—432no—
327C22—861no—
328C22—861no—
329C22—861no—
330C22—861no—
331C22—861no—
332C22C24×D4311yescharacteristic
333C4—23no—
334C4—21no—
335C4—23no—
336C4—21no—
337C4—23no—
338C4—432no—
339C4—863no—
340C4—863no—
341C4—863no—
342C4—861no—
343C4—863no—
344C4C2×#1062189816yes—
345C2—41no—
346C2—862no—
347C2—1721no—
348C2C22×#1062189811yescharacteristic
349C22−1+4×D4311yescharacteristic
350C22−1+4×D4311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D43
commutator, socleC22×C43C25
FittingC43×#519C2
hypercenter#519D43
hypocenterC43#3523
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ #519

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #14583627C2semidirect
2. #14583625C2semidirect
3. #14583617C2semidirect
4. #14583590C2semidirect
5. #14583588C2semidirect
6. Q8×D172C2semidirect
7. Q8×C22⋊D43C2semidirect
8. Q8×C22⋊D43C2semidirect
9. #14583511C2semidirect
10. #14583507C2semidirect
11. #14583453C2semidirect
13. #14583375C2semidirect
14. C4×Q8×D43C2semidirect
15. C43×#519C2semidirect
189. #519D43semidirect
191. C43#3523semidirect